**In a survey of 400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let p̂ be the fraction of survey respondents who preferred the incumbent.**

**Give an estimate of p.**

p̂ = the sample mean = 215 / 400 = 0.5375

This is also the estimate of the population mean, *p*.

**Calculate the Standard Error (SE) of the estimate, p̂.**

This is a Bernoulli sample. So, the sample variance for a Bernoulli sample is:

s_{x}^{2 }= p̂ * (1- ?p̂ )

s_{x}^{2 }= 0.5375*(1-0.5375)

s_{x}^{2 }= 0.2486

So, the SE of the estimated mean is:

_{x}^{2} / n)

SE = sqrt(0.2486 / 400)

SE = 0.0249

**What is the p-value for the test H_{0} : p = 0.5 vs. H_{1} : p != 0.5?**

The first thing to note is that, because the alternate hypothesis is !=, this is a two-sided hypothesis.

Next, compute the t-statistic assuming the null hypothesis:

t = ( (Sample Mean) – (Null Hypothesis) / SE )

t = (0.5375 – 0.5) / 0.0249

t = 1.506

The *p*-value for the two sided hypothesis is, therefore,

*p* = 2 * NORMDIST(-1.506, 0, 1, 1)

*p* = .1321

Note that we use the negative t-statistic in this calculation of area under the curve because we are interested in the area that is inconsistent with the null hypothesis.

**What is the p-value for the test H_{0} : p = 0.5 vs. H_{1} : p > 0.5?**

This time, we have a one-sided hypothesis test. So, the answer is half of what it was for the two sided test:

*p* = NORMDIST(-1.506, 0, 1, 1)

*p* = 0.066

The results of the one-sided hypothesis test differs from the two sided test because in the one-sided test, we are only interested in the case where the average is > the null hypothesis. In this case, we’re interested in the case where the incumbent may actually have less than 50% of the population support. In the two-sided case, we were interested in the case where the incumbent had support from less than 50% of the population as well as the case where the incumbent had more than 57.5% of the population’s support.

**Do the survey results contain statistically significant evidence that the incumbent was ahead of the challenger at the time of the survey?**

It depends on the chosen significance level. For significance levels > 0.066, there is statistical evidence that the incumbent is ahead.